Homework 1

Linear Regression

The data for this problem illustrate the relationship between the average monthly outdoor temperature (in degrees Fahrenheit) and the average monthly gas consumption (in therms) for a household.  Use the attached SAS output to answer the following questions.

1. Based on the first scatterplot, does it appear that it would be appropriate to fit a linear regression model to these data?  Why or why not?

1. A regression model was fit to these data using SAS.  Use the output from PROC REG to write down the prediction equation for this regression model.

1. What is the value of the slope for the model?  What does the slope tell you about the relationship between average temperature and average gas consumption?

1. Using information from the SAS output and table A.3 in Steele & Torrie, construct a 95% confidence interval for the slope.  How do you interpret this confidence interval in non-statistical terms?

1. Does it appear that there is a significant linear relationship between average temperature and average gas consumption?  Use the SAS output to conduct a hypothesis test to support your answer.   Be sure to state the hypotheses and report the t-statistic and p-value for your test.

1. Assuming that this model is correct, how much gas should we expect to use, on average, in a month when the average temperature was 45 degrees Fahrenheit?

1. Would it be appropriate to use this model to predict the average gas usage for a month when the average temperature was 85 degrees?  Why or why not?

1. Report the coefficient of determination (R2) for this model and explain (in non-statistical terms) what it tells you about the relationship between average temperature and average gas consumption.

1. Examine the plot of residuals vs. predicted values and residuals vs. time.  Does it appear that any of our assumptions have been violated or that there is need for further investigation before accepting this as our final regression model?  Explain.

The SAS System         13:35 Monday, January 20, 2003   9

Plot of avgas*avtemp.  Legend: A = 1 obs, B = 2 obs, etc.

12 ˆ

A            A

A

10 ˆ

A

A

A

8 ˆ                      A

A               A

avgas

6 ˆ

A

A

4 ˆ

A

A

2 ˆ                                                  A

A

A

0 ˆ

Šˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒ

25       30       35       40       45       50       55       60       65       70       75

avtemp

The SAS System         13:35 Monday, January 20, 2003   8

Obs    time    avtemp    avgas

1      0      29        8.9

2      1      30       11.6

3      2      31       10.7

4      3      37       11.6

5      4      48        7.5

6      5      57        3.5

7      6      68        1.5

8      8      71        0.8

9     10      53        1.9

10     11      40        5.0

11     12      39        7.3

12     13      29        9.3

13     14      36        9.7

14     15      37        7.9

15     16      46        5.8

16     17      56        3.2

The CORR Procedure

2  Variables:    avgas    avtemp

Pearson Correlation Coefficients, N = 16

Prob > |r| under H0: Rho=0

avgas        avtemp

avgas        1.00000      -0.90300

avtemp      -0.90300       1.00000

The REG Procedure

Model: MODEL1

Dependent Variable: avgas

Analysis of Variance

Sum of           Mean

Source                   DF        Squares         Square    F Value    Pr > F

Model                     1      161.51498      161.51498      61.85    <.0001

Error                    14       36.56252        2.61161

Corrected Total          15      198.07750

Root MSE              1.61605    R-Square     0.8154

Dependent Mean        6.63750    Adj R-Sq     0.8022

Coeff Var            24.34723

Parameter Estimates

Parameter       Standard

Variable     DF       Estimate          Error    t Value    Pr > |t|

Intercept     1       17.37277        1.42362      12.20      <.0001

avtemp        1       -0.24295        0.03089      -7.86      <.0001

The SAS System         13:35 Monday, January 20, 2003  12

Plot of res*pred.  Legend: A = 1 obs, B = 2 obs, etc.

4 ˆ

A

3 ˆ

2 ˆ

A

A

R

e  1 ˆ                                                          A

s                                                                      A

i       A    A

d

u

a

l  0 ˆ                         A

A

A                             A

A

-1 ˆ                                                                     A

A

-2 ˆ

A                    A

-3 ˆ

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0            2            4            6            8           10           12

Predicted Value of avgas

Plot of res*time.  Legend: A = 1 obs, B = 2 obs, etc.

4 ˆ

A

3 ˆ

2 ˆ

A

A

R

e  1 ˆ                                                                        A

s                A

i                                    A         A

d

u

a

l  0 ˆ                           A

A

A         A

A

-1 ˆ                                                                   A

A

-2 ˆ

A    A

-3 ˆ

Šƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒƒƒˆƒƒ

0    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17

time